Mandelbrot & Julia

Guide

Repeat the rule z → z² + c. Some starting points stay nearby; others escape. The Mandelbrot set maps the choices of c, while the other panel shows the corresponding Julia picture.

Try this: right-click in the parameter plane and move the mouse around the dark region’s boundary, or double-tap and drag on mobile. Watch the Julia picture change from connected shapes to scattered islands.

Controls

• Desktop: right-click a plot to switch between pan / zoom and point placement, independently for each plot. Drag to pan; in point placement mode the point follows the mouse. Scroll to zoom in either mode without moving the point.
• Mobile: drag with one finger to pan. To move a point, double-tap, keep the second tap held, and drag; lift your finger to finish. Use two fingers to pan and pinch to zoom. Adding a second finger cancels point placement, so pinching cannot leave the root or parameter accidentally moved.
• Orbit start: enable Show orbit, turn off its tracking/critical-point lock, and enable Move orbit start. Point placement in the dynamical plane then moves the orbit start instead of the parameter or root.
• Fullscreen: use Fullscreen to enter; use the browser’s Back gesture on mobile or Esc on desktop to leave. There is no exit button covering the plot.
• Keyboard: Tab to a plot, press M to switch modes, use the arrow keys to pan or move the point, + / − to zoom, and 0 to reset the view. Reset view also restores the initial pan and zoom.

Reading the picture

The dynamical plane fixes c and tries a different starting point at each pixel. The parameter plane tries a different c at each pixel, always starting at z = 0.

Colors encode escape time; very dark pixels have not escaped within the iteration limit. Staying dark for a finite number of iterations is not a proof that a point never escapes.

• Gold dot: the selected parameter c, shown at the same coordinates on both panels.
• Magenta dots: fixed points, where applying the rule leaves the point unchanged.
• Teal dot: the critical point z = 0.
• White path: an orbit when Show orbit is enabled. Turn off Lock to critical point and select Move orbit start to try another starting point.